Normal-Inverse-Gamma Distribution for Prior and Posterior

Creates an object containing the parameters of a Normal-inverse-gamma distribution. This is used for constructing the prior or storing the information of the posterior.

Usage

inz_dNIG(
  likelihood = NULL,
  mu = NULL,
  lambda = NULL,
  alpha = NULL,
  beta = NULL
)

Arguments

likelihood: The likelihood object of class "inz_lnorm".
mu: The normal mean parameter $\mu$. For regression, this is a vector of coefficient values.
lambda: The normal precision scaling parameter $\lambda$. For regression, this is a precision matrix $\Lambda$.
alpha: The inverse-gamma shape parameter $\alpha$.
beta: The inverse-gamma scale parameter $\beta$.

Value

Returns an object of class "inz_dNIG" (single variable/grouped) or "inz_dNIG_reg" (regression).

An object of class "inz_dNIG" or "inz_dNIG_reg" is a list which contains the following:

mu: the mean parameter $\mu$.
lambda: the precision scale parameter $\lambda$ or a precision matrix $\Lambda$.
V: the variance scale factor or a covariance matrix.
alpha: the shape parameter $\alpha$.
beta: the scale parameter $\beta$.

If the function is used to construct the prior, the likelihood is stored as an attribute.

Details

The Normal-Inverse-Gamma distribution is the conjugate prior for the Normal likelihood, when both the mean and the variance is unknown.

$\lambda > 0$
$\alpha > 0$
$\beta > 0$

Single Variable and Grouped Data: $$\sigma^2 \sim \Gamma^{-1}(\alpha, \beta)$$ $$\theta | \sigma^2 \sim \text{N}(\mu, \sigma^2V)$$ where $V=1/\lambda$, a variance scale factor.

For a single variable, $\mu$ and $\lambda$ are scalars.

For grouped data, $\mu$ and $\lambda$ are vectors where each value corresponds to a group.

Regression: $$\sigma^2 \sim \Gamma^{-1}(\alpha, \beta)$$ $$\beta | \sigma^2 \sim \text{N}(\mu, \sigma^2 \Lambda^{-1})$$

For regression, $\mu$ is a vector of length 2 and $\Lambda$ is a 2x2 precision matrix.

For prior use only: If no prior parameters are provided, uninformative default values are used for the prior:

mu = 0 (or vector of zeros)
lambda = 1 (or vector of ones, or identity matrix for regression)
alpha = 0.001
beta = 0.001

Examples

# Constructing the prior with the likelihood (default prior is used)
if (FALSE) { # \dontrun{
lik <- inz_lnorm(surf_data, Income)
inz_dNIG(likelihood = lik)

# Using a subjective prior
inz_dNIG(likelihood = lik, mu = 500, lambda = 10, alpha = 5, beta = 1)
} # }

# Regression example
if (FALSE) { # \dontrun{
reg_lik <- inz_lnorm(surf_data, Income, Hours)
inz_dNIG(likelihood = reg_lik)
} # }

# Example of inz_dNIG usage in the calculate_posterior function
inz_dNIG(mu = 572.4975, lambda = 201, alpha = 100.001, beta = 12118391)
#> $mu
#> [1] 572.4975
#> 
#> $lambda
#> [1] 201
#> 
#> $V
#> [1] 0.004975124
#> 
#> $alpha
#> [1] 100.001
#> 
#> $beta
#> [1] 12118391
#> 
#> attr(,"class")
#> [1] "inz_dNIG"